Metric envelope and simulation allowance

Suppose $(X,d)$ is a supplied metric space and $\mathrm{diam}B\le \Delta$ for every block. Induction in (eq:ancestors) gives

Indeed each backward step adds at most $\Delta$ to the distance. For externally declared tick duration $\delta >0$, the scalar envelope rate is $\Delta /\delta$. Neither $d$ nor $\delta$ is derived by this proof.

Use the conventions

These standard complete-channel norms include arbitrary reference systems Watrous.